© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:
K11n116
K11n116
K11n118
K11n118
K11n117
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   The Knot K11n117

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Acknowledgement

K11n117 as Morse Link
DrawMorseLink

PD Presentation: X4251 X10,3,11,4 X5,14,6,15 X7,19,8,18 X9,17,10,16 X2,11,3,12 X13,21,14,20 X15,22,16,1 X17,9,18,8 X19,13,20,12 X21,7,22,6

Gauss Code: {1, -6, 2, -1, -3, 11, -4, 9, -5, -2, 6, 10, -7, 3, -8, 5, -9, 4, -10, 7, -11, 8}

DT (Dowker-Thistlethwaite) Code: 4 10 -14 -18 -16 2 -20 -22 -8 -12 -6

Alexander Polynomial: - 3t-2 + 9t-1 - 11 + 9t - 3t2

Conway Polynomial: 1 - 3z2 - 3z4

Other knots with the same Alexander/Conway Polynomial: {1020, 10162, ...}

Determinant and Signature: {35, 2}

Jones Polynomial: q-3 - 2q-2 + 4q-1 - 5 + 6q - 6q2 + 5q3 - 4q4 + 2q5

Other knots (up to mirrors) with the same Jones Polynomial: {10138, ...}

A2 (sl(3)) Invariant: q-10 + 2q-4 + 1 - 2q4 - 2q8 + q10 + 2q16

HOMFLY-PT Polynomial: 2a-4 + 2a-4z2 - 3a-2 - 5a-2z2 - 2a-2z4 + 1 - z2 - z4 + a2 + a2z2

Kauffman Polynomial: 2a-6z2 - a-5z + 2a-5z3 + a-5z5 + 2a-4 - 7a-4z2 + 10a-4z4 - 4a-4z6 + a-4z8 + a-3z - 7a-3z3 + 9a-3z5 - 4a-3z7 + a-3z9 + 3a-2 - 14a-2z2 + 17a-2z4 - 11a-2z6 + 3a-2z8 + 2a-1z - 4a-1z3 + a-1z5 - 2a-1z7 + a-1z9 + 1 - z2 + 3z4 - 6z6 + 2z8 + 5az3 - 7az5 + 2az7 - a2 + 4a2z2 - 4a2z4 + a2z6

V2 and V3, the type 2 and 3 Vassiliev invariants: {-3, -2}

Khovanov Homology:
(The squares with yellow highlighting are those on the "critical diagonals", where j-2r=s+1 or j-2r=s+1, where s=2 is the signature of 11117. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)
  
trqj r = -4r = -3r = -2r = -1r = 0r = 1r = 2r = 3r = 4
j = 11        2
j = 9       2 
j = 7      32 
j = 5     32  
j = 3    33   
j = 1   34    
j = -1  12     
j = -3 13      
j = -5 1       
j = -71        


Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=    
<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=
Show[DrawMorseLink[Knot[11, NonAlternating, 117]]]
Out[2]=   
 -Graphics- 
In[3]:=
PD[Knot[11, NonAlternating, 117]]
Out[3]=   
PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[5, 14, 6, 15], X[7, 19, 8, 18], 
 
>   X[9, 17, 10, 16], X[2, 11, 3, 12], X[13, 21, 14, 20], X[15, 22, 16, 1], 
 
>   X[17, 9, 18, 8], X[19, 13, 20, 12], X[21, 7, 22, 6]]
In[4]:=
GaussCode[Knot[11, NonAlternating, 117]]
Out[4]=   
GaussCode[1, -6, 2, -1, -3, 11, -4, 9, -5, -2, 6, 10, -7, 3, -8, 5, -9, 4, -10, 
 
>   7, -11, 8]
In[5]:=
DTCode[Knot[11, NonAlternating, 117]]
Out[5]=   
DTCode[4, 10, -14, -18, -16, 2, -20, -22, -8, -12, -6]
In[6]:=
alex = Alexander[Knot[11, NonAlternating, 117]][t]
Out[6]=   
      3    9            2
-11 - -- + - + 9 t - 3 t
       2   t
      t
In[7]:=
Conway[Knot[11, NonAlternating, 117]][z]
Out[7]=   
       2      4
1 - 3 z  - 3 z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=   
{Knot[10, 20], Knot[10, 162], Knot[11, NonAlternating, 117]}
In[9]:=
{KnotDet[Knot[11, NonAlternating, 117]], KnotSignature[Knot[11, NonAlternating, 117]]}
Out[9]=   
{35, 2}
In[10]:=
J=Jones[Knot[11, NonAlternating, 117]][q]
Out[10]=   
      -3   2    4            2      3      4      5
-5 + q   - -- + - + 6 q - 6 q  + 5 q  - 4 q  + 2 q
            2   q
           q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=   
{Knot[10, 138], Knot[11, NonAlternating, 117]}
In[12]:=
A2Invariant[Knot[11, NonAlternating, 117]][q]
Out[12]=   
     -10   2       4      8    10      16
1 + q    + -- - 2 q  - 2 q  + q   + 2 q
            4
           q
In[13]:=
HOMFLYPT[Knot[11, NonAlternating, 117]][a, z]
Out[13]=   
                           2      2                   4
    2    3     2    2   2 z    5 z     2  2    4   2 z
1 + -- - -- + a  - z  + ---- - ---- + a  z  - z  - ----
     4    2               4      2                   2
    a    a               a      a                   a
In[14]:=
Kauffman[Knot[11, NonAlternating, 117]][a, z]
Out[14]=   
                                           2      2       2                3
    2    3     2   z    z    2 z    2   2 z    7 z    14 z       2  2   2 z
1 + -- + -- - a  - -- + -- + --- - z  + ---- - ---- - ----- + 4 a  z  + ---- - 
     4    2         5    3    a           6      4      2                 5
    a    a         a    a                a      a      a                 a
 
       3      3                       4       4              5      5    5
    7 z    4 z         3      4   10 z    17 z       2  4   z    9 z    z
>   ---- - ---- + 5 a z  + 3 z  + ----- + ----- - 4 a  z  + -- + ---- + -- - 
      3     a                       4       2                5     3    a
     a                             a       a                a     a
 
                       6       6              7      7                    8
         5      6   4 z    11 z     2  6   4 z    2 z         7      8   z
>   7 a z  - 6 z  - ---- - ----- + a  z  - ---- - ---- + 2 a z  + 2 z  + -- + 
                      4      2               3     a                      4
                     a      a               a                            a
 
       8    9    9
    3 z    z    z
>   ---- + -- + --
      2     3   a
     a     a
In[15]:=
{Vassiliev[2][Knot[11, NonAlternating, 117]], Vassiliev[3][Knot[11, NonAlternating, 117]]}
Out[15]=   
{-3, -2}
In[16]:=
Kh[Knot[11, NonAlternating, 117]][q, t]
Out[16]=   
         3     1       1       1       3      1      2    3 q      3
4 q + 3 q  + ----- + ----- + ----- + ----- + ---- + --- + --- + 3 q  t + 
              7  4    5  3    3  3    3  2      2   q t    t
             q  t    q  t    q  t    q  t    q t
 
       5        5  2      7  2      7  3      9  3      11  4
>   3 q  t + 2 q  t  + 3 q  t  + 2 q  t  + 2 q  t  + 2 q   t


Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n117
K11n116
K11n116
K11n118
K11n118